Question: Simplify the following expression: $\dfrac{40x}{120x}$ You can assume $x \neq 0$.
Answer: $ \dfrac{40x}{120x} = \dfrac{40}{120} \cdot \dfrac{x}{x} $ To simplify $\frac{40}{120}$ , find the greatest common factor (GCD) of $40$ and $120$ $40 = 2 \cdot 2 \cdot 2 \cdot 5$ $120 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$ $ \mbox{GCD}(40, 120) = 2 \cdot 2 \cdot 2 \cdot 5 = 40 $ $ \dfrac{40}{120} \cdot \dfrac{x}{x} = \dfrac{40 \cdot 1}{40 \cdot 3} \cdot \dfrac{x}{x} $ $\phantom{ \dfrac{40}{120} \cdot \dfrac{1}{1}} = \dfrac{1}{3} \cdot \dfrac{x}{x} $ $ \dfrac{x}{x} = 1 $ $ \dfrac{1}{3} \cdot 1 = \dfrac{1}{3} $